a r X i v :c o n d -m a t /0204638v 2 [c o n d -m a t .s o f t ] 1 A u g 2002
Dynamics of collapsing and exploding Bose-Einstein condensed vortex state
Sadhan K.Adhikari
Instituto de F′?sica Te′o rica,Universidade Estadual Paulista,
01.405-900S?a o Paulo,S?a o Paulo,Brazil
(Dated:February 1,2008)
Using the time-dependent mean-?eld Gross-Pitaevskii equation we study the dynamics of small
repulsive Bose-Einstein condensed vortex states of 85Rb atoms in a cylindrical trap with low angular momentum ˉh L per atom (L ≤6),when the atomic interaction is suddenly turned attractive by manipulating the external magnetic ?eld near a Feshbach resonance.Consequently,the condensate collapses and ejects atoms via explosion and a remnant condensate with a smaller number of atoms emerges that survives for a long time.Detail of this collapse and explosion is compared critically with a similar experiment performed with zero angular momentum (L =0).Suggestion for future experiment with vortex state is made.
PACS numbers:03.75.Fi
I.INTRODUCTION
Recent observation [1,2]of Bose-Einstein condensates (BEC)of dilute trapped bosonic atoms with repulsive interaction at ultra-low temperature has intensi?ed the-oretical and experimental studies on various aspects of the condensate [3].One fascinating feature is the obser-vation of quantized vortices [4]and vortex lattice [5,6]in the condensate as this is intrinsically related to the existence of super?uidity.Another interesting feature is the formation of a stable condensate composed of a ?nite number of attractive atoms less than a critical number N cr [2].The third noteworthy feature is the observation of Feshbach resonances in 23Na [7],85Rb [8]and Cs [9]atoms,as in the presence of such resonances the e?ective atomic interaction can be varied in a controlled fashion by an external (background)magnetic ?eld [10].
For super?uid 4He(II)in a rotating container,no mo-tion of the ?uid is observed below a critical rotational frequency.Above this frequency quantized vortices ap-pear in 4He(II)manifesting its super?uidity.However,because of the strong interaction,the theoretical de-scription of this system is not easy.Quantized vor-tices have been observed [4,5,6]in trapped BEC and can be generated in theoretical mean-?eld models [11,12,13,14,15,16,17]based on the Gross-Pitaevskii (GP)[18]equation.Di?erent ways for generating vortices in a BEC have been suggested [13],e.g.,via spontaneous formation in evaporative cooling [14],via controlled exci-tation to an excited state [15],by stirring a BEC using a laser with an angular frequency above a critical value [12],or by the rotation of an axially symmetric trap with an angular frequency above a similar critical value [16].In contrast to liquid 4He(II),a trapped BEC of small size is dilute and weakly interacting,which makes a mean-?eld analysis appropriate.
The observation of a condensate of attractive 7Li atoms and the subsequent measurement of the critical number N cr [2]is in good agreement with the mean ?eld analyses in a spherically symmetric trap [19],although the agree-
ment is not as good in the case of 85Rb [20]atoms in an axially symmetric trap [21].If the number of atoms can somehow be increased beyond this critical number,due to interatomic attraction the condensate collapses emit-ting atoms until the number of atoms is reduced below N cr .With a supply of atoms from an external source the condensate grows again beyond N cr and a sequence of collapses has been observed in 7Li by Gerton et al at Rice [2],where the number of atoms remain close to N cr and the collapse is driven by a stochastic process.Recently,a more challenging experiment was per-formed by Donley et al.[22]at JILA on a BEC of 85Rb atoms [8]in an axially symmetric trap,where they var-ied the interatomic interaction by an external magnetic ?eld near a Feshbach resonance [10].Consequently,they were able to change the sign of the atomic scattering length,thus transforming a repulsive condensate of 85Rb atoms into a collapsing and highly explosive attractive condensate and studied the dynamics of the same [22].Immediately after the jump in the scattering length,one has a highly unstable BEC,where the number of atoms could be much larger than N cr .Donley et al.[22]have provided a quantitative estimate of the explosion by mea-suring the number of atoms remaining in the BEC as a function of time until an equilibrium is reached.Because this phenomenon of emission of a very large number of atoms in a small interval of time is reminiscent of an ex-plosion and looks very much like a tiny supernova,or exploding star,the researchers dubbed it a “Bosenova”.The essential aspects of the above experiments at Rice by Gerton et al [2]and at JILA by Donley et al [22]have been theoretically described by a variety of authors us-ing the GP equation [18].The theoretical analyses have not only produced the time-independent results,such as,the critical number N cr [19,21],but also time-dependent results,such as,the variation of number of atoms of the BEC during collapse and explosion [23,24,25,26,27,28],both in reasonable agreement with experiment.This con-solidates the use of the mean-?eld GP equation in de-scribing the dynamics of collapsing and exploding BEC
2
of small to medium size.These BEC’s composed of sev-eral thousand atoms can be considered dilute and weakly interacting and hence amenable to mean-?eld treatment. Motivated by the above success,using the GP equation we propose the numerical simulation of the dynamics of a rotating Bosenova with a single vortex composed of a small number(several thousands)of85Rb atoms as in the experiment at JILA[22].We consider a single vortex state[4,11,21],as appropriate for small condensates,op-posed to vortex lattice for large condensates[5,6,17].A comparison of the present results with future experiment will provide a more stringent test for the mean-?eld GP equation.
In this paper we perform a mean-?eld analysis based on the time-dependent GP equation to understand the collapse and explosion of the attractive vortex state of 85Rb atoms in an axially symmetric trap.To account
for the loss of atoms from the strongly attractive con-densate we include an absorptive nonlinear three-body recombination term in the GP equation.The three-body recombination rate we use in numerical simulation is the same as used in a similar study with BEC’s with zero an-gular momentum[27]and is in agreement with previous experimental measurement[29]and theoretical calcula-tion[30].In the present investigation we consider the complete numerical solution of the mean-?eld GP equa-tion for an axially symmetric trap as in the experiment at JILA[22].As in the experiment at JILA with non-rotating condensates,we?nd that,also in the case of rotating vortex states,a large number of atoms could be emitted in a small interval of time and one could have a Bosenova-type explosion.
Throughout the present numerical simulation we make the assumption that the axial symmetry of the system is maintained.For small values of nonlinearity a dynamical quadrupole instability may cause an attractive BEC vor-tex state to split into two pieces that rotate around the axial direction[31].These pieces may unite to recover the original vortex and this split-merge cycle repeats.Simi-lar instability is known to exist for BEC in a toroidal trap [32].Clearly,a full three-dimensional calculation of the collapsing phenomena taking into consideration the e?ect of the splitting of vortex states seems to be practically impossible at present and will be a welcome future work. In view of this here we present an axisymmetric model of the same which is expected to provide the essentials of the collapse dynamics of the vortex states.
In Sec.II we present the theoretical model and the nu-merical method for its solution.In Sec,III we present our results.Finally,in Sec.IV we present a brief discussion and concluding remarks.
II.THE GROSS-PITAEVSKII EQUATION
A.Theoretical Model
The time-dependent Bose-Einstein condensate wave functionΨ(r;τ)at position r and timeτallowing for atomic loss may be described by the following mean-?eld nonlinear GP equation[3,18]
?iˉh?2m+V(r)+G N|Ψ(r;τ)|2?iˉh
2
mω2(r2+λ2z2)whereωis the an-gular frequency in the radial direction r andλωthat in the axial direction z,withλthe aspect ratio.We are using the cylindrical coordinate system r≡(r,θ,z)with θthe azimuthal angle.The normalization condition of the wave function is d r|Ψ(r;τ)|2=1.
The GP equation can easily accommodate quantized vortex states with rotational motion of the BEC around the z axis.In such a vortex the atoms?ow with tan-gential velocity Lˉh/(mr)such that each atom has quan-tized angular momentum Lˉh along the z axis.This cor-responds to an angular dependence of
Ψ(r,τ)=ψ(r,z,τ)exp(iLθ)(2.2) of the wave function,where exp(iLθ)are the circular harmonics in two dimensions.
Now transforming to dimensionless variables de?ned by x=
√2z/l,t=τω,l≡
x≡
√
?t?
?2
x
?
?y2
+
L2?1
4 x2+λ2y2 +κn ?(x,y;t)x 4 ?(x,y;t)=0,(2.4) where n=Na/l,κ=8
√
3 found that for negative a,K3increases rapidly as|a|n,
where the theoretical study favors n=2and we represent
this variation via this quadratic dependence.This makes
the parameterξabove a constant
[27]
for an experimental
set up with?xed l andωand in the present study we
employ a constantξ.
The normalization condition of the wave function for
K3=0is
N norm≡2π ∞0dx
N norm
∞0dx ∞?∞dy|?(x,y;t)|2x,(2.6)
y2rms=
2π
x ∞?∞dy|?(x,y;t)|2y2.(2.7)
B.Calculational Detail
We solve the GP equation(2.4)numerically using
a split-step time-iteration method using the Crank-
Nicholson discretization scheme[21,27,34,35,36].We
discretize the GP equation with time step0.001and space
step0.1spanning x from0to15and y from?35to35.
It is now appropriate to calculate the parameters of the
present dimensionless GP equation(2.4)corresponding
to the experiment at JILA for L=0[22].As in that
experiment we take the radial and axial trap frequencies
to beνradial=17.5Hz andνaxial=6.8Hz,respectively,
leading toλ=0.389.The harmonic oscillator length l of
85Rb atoms forω=2π×17.5Hz and m≈79176MeV
is l=
22L+3π3(|L|!)2 1/4x1+|L|e?(x2+λy2)/4(2.8)
for n=0.In the course of above time iteration the
nonlinear parameter n was increased by steps of0.0001
until its?nal value is attained.Then during an interval
of time0.1ms the scattering length was ramped to a=
a collapse.The absorptive termξwas set equal to zero
throughout above time iteration.
The?nal condensate is strongly attractive and un-
stable and undergoes a sequence of collapse and ex-
plosion.In our numerical simulation with L=0
we consider a set of di?erent values of a collapse(=
?263a0,?100a0,?30a0,?20a0,etc.)as well as N0=
6000.
For the simulation of collapse and explosion a nonzero
value ofξ(=2)is chosen for di?erent a initial,a collapse,
and N0as in Ref.[27]and the time-evolution of the
GP equation is continued.This value ofξreproduced
the essentials of the experiment at JILA[22]reasonably
well for L=0as well as produced[27]a K3in reasonable
agreement with a previous experiment(K3=4.24×10?25
cm6/s)[29]and theoretical calculation(K3=6.7×10?25
cm6/s)[30]for a=?370a0.In particular we use K3=
9×10?25cm6/s for a=?370a0.For smaller values
of|a|,the K3values are scaled down using the relation
K3∝a2.
III.NUMERICAL RESULT
The numerical simulation using Eq.(2.4)with a
nonzeroξas described above immediately yields the re-
maining number of atoms in the condensate after the
jump in scattering length.The remaining number of
atoms vs.time is plotted in Fig.1(a)for a initial=7a0,
a collapse=?30a0and?263a0,N0=16000,and
L=0,1,2,4,and6.In Fig.1(b)the same results for
N0=6000are plotted.In this?gure we also plot some
results of experiment at JILA for L=0[22,27].These
experimental results are in agreement with the simulation
for L=0.In Fig.2(a)we plot the particle loss curves
for N0=6000,a initial=7a0and di?erent a collapse for
L=0.In Fig.2(b)we plot the same for L=1.
From Figs.1and2we?nd that for a?xed N0,for
a su?ciently small L or a su?ciently large|a collapse|,
there could be collapse and explosion during a relatively
short interval of time(called decay time)with the loss of a
large fraction of the atoms.However,there is no collapse
for a large enough L or a small enough|a collapse|.For
example,for N0=6000in Fig.1(b)there is no collapse
for L>2for|a collapse|=30a0and in Fig.2(a)there
is no collapse for|a collapse|<5a0for L=0.
In the experiment at JILA[22]for L=0it was
observed that the strongly attractive condensate after
preparation remains stable with a constant number of
atoms for an interval of time t collapse,called collapse
time.This behavior is physically expected for medium
to small values of|a collapse|(<50a0).Immediately af-
ter the jump in the scattering length to a negative value,
the attractive condensate shrinks in size during t collapse,
until the central density increases to a maximum.Then
the absorptive three-body term takes full control to ini-
tiate the explosion that last for few milliseconds.Conse-
quently,the number of atoms remains constant for time
t also show this behavior for|a collapse|=30a0.However, 4 for larger|a collapse|(=263a0),the atomic attraction is very strong and the central density increases to a maxi-mum quickly to start the explosion and t collapse is close to zero.In Figs.2(a)and(b)we see the dependence of particle loss and t collapse on|a collapse|for N0=6000, a initial=7a0,and L=0and L=1,respectively.From number of particles drop sharply during a small interval of time called decay time(few milliseconds),which means that the condensate emits a large number particles in an explosive fashion.This emission of particles is termed explosion. After a sequence of collapse and explosion,for L= 0Donley et al.[22]observed a“remnant”condensate 5 place is small and of the order of few milliseconds.The decay time for vortex states(L=0)is smaller than for nonrotating condensates(L=0). We studied the time evolution of the condensate for larger times.In Fig.3we plot the loss curves for N0=16000,a initial=7a0,a collapse=?30a0,and L=0,1,2,4,6at larger times.The BEC continues to lose atoms if left for a long time but at a rate much slower than during the?rst explosion that we call pri-mary.However,in the experiment at JILA Donley et al observed that a remnant condensate containing a fraction of the atoms survived with nearly constant number for more than one second.One possible reason for this dis-crepancy could be the following.In the actual experiment at JILA a major portion of the emitted atoms,called the burst atoms,remain trapped and oscillate around the central remnant.The presence of the burst atoms make the measurement of the number of atoms in the rem-nant a di?cult task[37].Some of these burst atoms may also rejoin the remnant to compensate for the three-body loss at large times.Such an e?ect is not included in the present model,which,hence,presents a larger loss for the remnant compared to experiment. We also observe an interesting phenomenon in Fig.3, e.g.,the occurrence of smaller secondary and tertiary ex-plosions after the primary one observed for small times.plosions are separated by a large interval of time.We also see much weaker explosion(s)in the course of time evolution in Fig.3,where the particle number varies in small steps.These explosions could be termed tertiary with the loss of few hundred atoms.It might be inter-esting to see if such secondary and tertiary explosion(s) could be observed experimentally. Donley et al.[22]provided a quantitative measure-ment of the variation of collapse time t collapse with the ?nal scattering length a collapse for a given a initial=0, N0=6000,and L=0.We calculated this variation using our model for L=0,1and2.The t collapse vs. |a collapse|/a0plots for L=0,1,2are exhibited in Fig. 4and compared with experimental data for L=0[22] as well as with another calculation using the mean-?eld GP equation in an axially symmetric trap for L=0[26]. We see t collapse decreases with|a collapse|/a0starting from an in?nite value at|a collapse|=a cr,where a cr is the minimum value of|a collapse|that leads to col-lapse and explosion.The critical value a cr increases with L and so does t collapse for a?xed|a collapse|.For a given N0,a critical value of n≡n cr can be de?ned via n cr≡N0a cr/l.A necessary condition for collapse is N0|a collapse|/l>n cr[21].The value of n cr for a speci?c 6 case is calculated as in Ref. [21].Consequently,